McClellan Transformation Research Articles

Design of two‐channel perfect reconstruction QMF

AbstractThis paper proposes a design method for the FIR two‐channel perfect quadrature mirror filter (QMF) with the linear phase. The two‐channel perfect QMF can be designed by Vetterli's method, where a system of equations representing the condition for the perfect reconstruction of the signal is solved. The filter designed by this method, however, does not, in general, have good frequency characteristics.This paper presents the design for the two‐channel QMF with a perfect reconstruction and a good amplitude characteristic. The method is based on Vetterli's method, and a constraint to approximate the amplitude characteristic in the frequency domain is added to the condition of perfect reconstruction in time domain. Then Remez' algorithm is applied to the derived system of equations.The construction of the two‐channel perfect QMF, when the coefficient is quantized, also is discussed. Furthermore, a method is shown in which the two‐dimensional (2‐D) perfect QMF is designed by applying the McClellan transformation to the obtained 1‐D QMF. The condition for the McClellan transformation for the perfect reconstruction is derived.

Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables

An approach to designing multidimensional linear-phase FIR diamond subband filters having the perfect reconstruction property is presented. It is based on a transformation of variables technique and is equivalent to the generalized McClellan transformation. Methods for designing a whole class of transformation are given. The approach consists of two parts; design of the transformation and design of the 1-D filters. The use of Lagrange halfband filters to design the 1-D filters is discussed. The modification of a particular Lagrange halfband filter which gives a pair of simple 1-D filters that are almost similar to each other in their frequency characteristics but still form a perfect reconstruction pair is presented. The design technique is extended to other types of two-channel sampling lattice and subband shapes, in particular, the parallelogram and the diagonally quadrant subband cases. Several numerical design examples are presented to illustrate the flexibility of the design method.

3-D migration via McClellan transformations on hexagonal grids

Two‐dimensional (2-D) spatially varying convolutional filters for three‐dimensional (3-D) depth extrapolation are efficiently designed and implemented by using McClellan transformations. Although efficient, McClellan transformations only approximate circularly symmetric depth extrapolation filters. The accuracy of the extrapolation filters can be increased only at an increase in the computational cost of implementing those filters. By applying McClellan transformations on a hexagonal sampling grid, the accuracy of the extrapolation filters can be increased at a reduction in cost.

Two-dimensional adaptive digital filters with reduced computational complexity

A 2-D adaptive filter structure based on the McClellan transformation design technique for 2-D FIR filters is proposed as a means of achieving improved performance in 2-D adaptive filters. performance is compared to that of a direct form 2-D LMS structure in terms of learning characteristics and computational efficiency. It is first shown that if the transformation structure is constrained by a priori knowledge of contour shapes in the frequency domain, the 2-D adaptive algorithm results in greatly reduced computational requirements and more rapid learning characteristics, as compared to the direct form. The transformation filter is then generalized by including the contour parameters in the adaptive parameter set to eliminate the constraints on the frequency domain contours. Finally, it is shown how an orthogonal transformation can be incorporated to improve the convergence rate of the constrained transformation structure.

Algorithms and systolic architectures for multidimensional adaptive filtering via McClellan transformations

Algorithms are developed simultaneously with systolic architectures for multidimensional adaptive filtering Because of the extremely high data rate required for real-time video processing, there is a strong motivation to limit the size of any adaptation problem. Combining the McClellan transformations with systolic arrays to adapt and implement the least-squares filter yields a novel solution to the problem of adapting a large zero-phase finite impulse response (FIR) multidimensional filter, having arbitrary directional biases, with only a few parameters. These filters can be adapted abruptly on a block-by-block basis without causing blocking effects. After developing a basic processing element for a systolic array realization of the Chebyshev structure for the McClellan transformation, it is shown that for a given 2-D transformation function, the adaptation of the 1-D prototype filter becomes a small multichannel adaptation problem similar to adaptive array problems. A similar approach is also taken in developing algorithms to adapt the 2-D transformation function. >

Design of 2-D FIR filter possessing purely imaginary frequency response by the transformation method

Proposes a novel method for the design of a 2-D FIR filter possessing a purely imaginary frequency response involving the McClellan transformation and the transformation method proposed by Y.L. Tai and T.P. Lin (1989). The McClellan transformation can be used only for the design of symmetric 2-D FIR filters. The proposed method can be used not only for the design of 2-D FIR filters possessing a purely imaginary frequency response, but also for the design of symmetric 2-D FIR filters (possessing a purely real frequency response) with the same structures that are proposed for the McClellan transformation method with some modifications. >

To: “3-D depth migration via McClellan transformations” by Dave Hale (GEOPHYSICS, 56, 1778–1785, November 1991).

On p. 1783 of this paper, Figures 9 and 10 were switched and should be reversed to agree with the captions.

On the cascade realization of 2-D FIR filters designed by McClellan transformation

Multidimensional finite impulse response (FIR) filters designed by the McClellan transform can be implemented efficiently by the direct transformed structure, transpose direct transformed structure, cascade structure, Chebyshev structure, and reversed Chebyshev structure. Peak scaling and section reconfiguration are provided here to modify the cascade structure realization. The modifications result in a reduction in the output roundoff noise power and the number of operations required.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of zero-phase spherically symmetric N-dimensional IIR digital filters

The paper presents a new method for designing zero-phase N-dimensional IIR digital filters satisfying given amplitude design specifications. The method is based on applying the N-dimensional extension of McClellan transformation to the square of a one-dimensional IIR digital filter design recently proposed by Charalambous. The resulting N-dimensional digital filter is of zero-phase and hyperspherically symmetric but unstable. Stabilisation of the filter is achieved by dividing it into a product of 2N filters, each of which is stable if realised through a recursion in the suitable quadrant. The stabilisation procedure uses the properties of the complex cepstrum. Three examples are introduced to show the performance and validity of the design method presented. The designs obtained are also tested in image processing applications.

Direct synthesis of near-optimum difference patterns for planar arrays

A technique for the synthesis of planar arrays with difference patterns, which makes use of a McClellan transformation, trigonometric addition formulas and linear array synthesis methods, is described. >

SCALED AND SCALING-FREE McCLELLAN TRANSFORMATIONS FOR THE DESIGN OF 2-D DIGITAL FILTERS

In this paper, the scaling problem of McClellan transformation used for the design 2-D digital filters with elliptical and circular cutoff contours is critically analyzed. Several new and extremely simple formulas for calculating the maximum and minimum values of the transformation function are presented. Using these extreme values, simple formulas for scaling factors and scaled McClellan transformations are derived for the various cases of each type of contour. Using the necessary and sufficient conditions for a scaling-free transformation, the scaling-free McClellan transformations are also derived. It is shown that the number of independent coefficients in them to be optimized varies from 0 to 2 depending on the case, compared to 3 in the original transformation. They are very useful in deriving analytical expressions for the transformation coefficients, and the 2-D filters designed by employing them have a smaller number of multiplications per output sample and hence, attractive for real-time applications. A new scaling formula is presented and some of its properties and effects on the derived formulas are discussed. The major advantage of this new formula is that from the unscaled transformation coefficients for a given type of 2-D filter, we can generate the scaled transformation coefficients for a different type of 2-D filter. Some examples are presented to demonstrate the usefulness of the derived formulas.

Transformation-based synthesis technique for planar arrays with contoured beams

A transformation-based technique for the synthesis of planar arrays with contoured beams in terms of two decoupled simpler subproblems is described. The method makes use of the McClellan transformation and linear array synthesis methods.

Equiripple approximation design using first order McClellan transformation

This paper describes a technique, using the Remez algorithm, to determine the coefficients of the first-order McClellan transformation so that the cut-off frequency of the prototype one-dimensional filter maps onto a contour in the two-dimensional frequency plane which approximates a specified contour in equiripple sense. The type of contours considered are the passband boundaries of circularly symmetric, elliptically symmetric and α°-fan filters. The initial estimate of the transformation coefficients and the cut-off frequency of the one-dimensional filter is obtained analytically. Some of these parameters are then refined using Remez algorithm. Since the initial estimate of the parameters is good, the Remez algorithm converges very rapidly.

Optimal McClellan transformation and its application to 2-D FIR digial filter design

A new method to choose the coefficients of the McClellan transformation is developed. The coefficients are obtained by using both analytic and nonlinear optimization approaches. The transformation using the proposed method has much better approximation performance for circular symmetry than the existing methods, especially when the cutoff frequency of the 2-D filter is very large. The proposed method also applies to choose the transformation coefficients to approximate elliptic contours. Two design examples of 2-D FIR digital filters demonstrate the good performance of the proposed method.

A method of designing FIR fan filters based on McClellan transformations

AbstractThis paper describes a method of designing the fan filter with a steep cutoff characteristic, using McClellan transformation. The conventional design by McClellan transformation has a problem in that the cutoff region of the fan filter is broad due to the restriction to a lower order of the subfilter employed.From such a viewpoint, this paper proposes a systematic design method for the subfilter with arbitrary order. First, it is pointed out that the ideal response of the subfilter can be represented in a functional form. Then the error function is defined in regard to the ideal response, and the design is shown for the subfilter where the filter coefficients minimize the error function. Finally, a design example is discussed, and it is shown that a fan filter with a steeper cutoff characteristic than in the conventional method can be designed by the proposed method.

3-D depth migration via McClellan transformations

Three‐dimensional seismic wavefields may be extrapolated in depth, one frequency at a time, by two‐dimensional convolution with a circularly symmetric, frequency‐ and velocity‐dependent filter. This depth extrapolation, performed for each frequency independently, lies at the heart of 3-D finite‐difference depth migration. The computational efficiency of 3-D depth migration depends directly on the efficiency of this depth extrapolation. McClellan transformations provide an efficient method for both designing and implementing two‐dimensional digital filters that have a particular form of symmetry, such as the circularly symmetric depth extrapolation filters used in 3-D depth migration. Given the coefficients of one‐dimensional, frequency‐ and velocity‐dependent filters used to accomplish 2-D depth migration, McClellan transformations lead to a simple and efficient algorithm for 3-D depth migration. 3-D depth migration via McClellan transformations is simple because the coefficients of two‐dimensional depth extrapolation filters are never explicitly computed or stored; only the coefficients of the corresponding one‐dimensional filter are required. The algorithm is computationally efficient because the cost of applying the two‐dimensional extrapolation filter via McClellan transformations increases only linearly with the number of coefficients N in the corresponding one‐dimensional filter. This efficiency is not intuitively obvious, because the cost of convolution with a two‐dimensional filter is generally proportional to [Formula: see text]. Computational efficiency is particularly important for 3-D depth migration, for which long extrapolation filters (large N) may be required for accurate imaging of steep reflectors.

Design and implementation of circularly and spherically symmetric multidimensional digital filters

A practical design and implementation approach for two- and three-dimensional digital filters with symmetrical magnitude response is presented. The proposed filters have not only good circularly or spherically symmetric magnitude response, but also nearly linear phase response as well as several desirable stability properties. The denominator of the corresponding filter transfer function is separable in each dimension, while the numerator can be obtained by a one-dimensional approximation and the generalized McClellan transformation. The filters can be implemented by using either conventional one-dimensional second-order digital filters or wave digital filters. Thus, they are especially suitable for real-time applications. When strict linearity of the phase response is required, a simple approach can be used for phase equalization.

Circularly symmetric two-dimensional multiplierless FIR digital filter design using an enhanced McClellan transformation

A new analytical approach to the determination of the coefficients of the 1st-order McClellan transformation is presented. The resultant 1st-order expression can then be used to design 2-D (2-dimensional) lowpass/highpass, bandpass/bandstop and multiband digital filters from the corresponding 1-D digital filters with a high accuracy in the circularity of magnitude response along the passband cut-off frequency. On comparing the results with those of the original 1st-order McClellan transformation, it is found that the present analytical approach does improve the effectiveness of the original McClellan transformation for 1-D to 2-D digital filter transformation. Based on the resultant 1st-order expression, the realisation of 2-D FIR structures with sum of powers-of-two coefficients is also presented.

Reversed Chebyshev implementation of McClellan transform and its roundoff error

A new structure, called the reversed Chebyshev structure, for the fixed point implementation of the FIR filters designed with the McClellan transformation is presented. It shows the best roundoff noise performance among the existing ones for both rounding and truncation operations. The incremental rate of the upper bound on the variance of the roundoff noise, as the size of the 2-D filters increases, is very small. The reversed Chebyshev structure is rooted from our calculation of the upper bound on the variance of the output roundoff noise of 2-D FIR filters designed with the McClellan transformation. This upper bound is independent of the transformation filters and can be easily calculated to estimate the number of bits needed for the desired accuracy in the fixed point implementation.

Design of elliptically symmetric two-dimensional FIR filters using the McClellan transformation

An analytic technique for choosing the coefficients of the first-order McClellan transformation for the design of elliptically symmetric two-dimensional FIR filters is described. It is found that the results obtained by this analytic technique are nearly identical to those obtained by an optimization procedure reported earlier. The number of multiplications per output sample required to implement these filters is shown to be small.